A new logarithmic transformation for generating flexible probability distributions: Statistical properties, inference, and applications

Aadil Ahmad Mir; A. A. Bhat; S. P. Ahmad; Badr S. Alnssyan; Abdelaziz Alsubie; Aadil Ahmad Mir; A. A. Bhat; S. P. Ahmad; Badr S. Alnssyan; Abdelaziz Alsubie

doi:10.3934/math.20261343

AIMS Mathematics

2026, Volume 11, Issue 10: 34417-34467. doi: 10.3934/math.20261343

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Research article Special Issues

A new logarithmic transformation for generating flexible probability distributions: Statistical properties, inference, and applications

1.
Department of Statistics, University of Kashmir, Srinagar 190006, India
2.
Department of Mathematical Sciences, Islamic University of Science and Technology, Awantipora 192122, India
3.
Department of Management Information Systems, College of Business and Economics, Qassim University, Buraydah 52571, Saudi Arabia
4.
Department of Basic Sciences, College of Science and Theoretical Studies, Saudi Electronic University, Riyadh, Saudi Arabia

Received: 12 July 2026 Revised: 09 September 2026 Accepted: 10 October 2026 Published: 22 October 2026
MSC : 60E05, 62E10, 62N05, 62F10, 94A17

Real-world lifetime data often exhibit complex hazard-rate patterns that conventional models, such as the Weibull distribution, may not adequately capture. This study introduces a new three-parameter New Log-G (NLG) family of distributions, constructed through a logarithmic transformation of a baseline cumulative distribution function with an additional shape-controlling parameter $ \alpha $. Using the Weibull distribution as the baseline, we derive the New Log-Weibull (NLW) distribution, which accommodates increasing, decreasing, bathtub-shaped, and unimodal hazard-rate patterns. The principal theoretical properties of the proposed family are derived, including distributional functions, moments, entropy measures, and residual lifetime characteristics. Six classical estimation methods are evaluated through Monte Carlo simulations across varying sample sizes and parameter configurations. The practical usefulness of the NLW distribution is demonstrated using three real datasets on radiation susceptibility, glass fiber tensile strength, and COVID-19 mortality. Across these applications, the NLW model consistently outperforms seven established competing distributions in terms of goodness of fit. These results demonstrate the flexibility and practical utility of the proposed NLG family as a general framework for extending lifetime distributions, with the NLW model offering broad applicability in reliability engineering, materials science, and epidemiological modeling.
- New Log-Weibull distribution,
- reliability assessment,
- COVID-19 mortality modeling,
- classical estimation methods,
- simulation,
- goodness-of-fit evaluation
Citation: Aadil Ahmad Mir, A. A. Bhat, S. P. Ahmad, Badr S. Alnssyan, Abdelaziz Alsubie. A new logarithmic transformation for generating flexible probability distributions: Statistical properties, inference, and applications[J]. AIMS Mathematics, 2026, 11(10): 34417-34467. doi: 10.3934/math.20261343

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Abstract

Real-world lifetime data often exhibit complex hazard-rate patterns that conventional models, such as the Weibull distribution, may not adequately capture. This study introduces a new three-parameter New Log-G (NLG) family of distributions, constructed through a logarithmic transformation of a baseline cumulative distribution function with an additional shape-controlling parameter $ \alpha $. Using the Weibull distribution as the baseline, we derive the New Log-Weibull (NLW) distribution, which accommodates increasing, decreasing, bathtub-shaped, and unimodal hazard-rate patterns. The principal theoretical properties of the proposed family are derived, including distributional functions, moments, entropy measures, and residual lifetime characteristics. Six classical estimation methods are evaluated through Monte Carlo simulations across varying sample sizes and parameter configurations. The practical usefulness of the NLW distribution is demonstrated using three real datasets on radiation susceptibility, glass fiber tensile strength, and COVID-19 mortality. Across these applications, the NLW model consistently outperforms seven established competing distributions in terms of goodness of fit. These results demonstrate the flexibility and practical utility of the proposed NLG family as a general framework for extending lifetime distributions, with the NLW model offering broad applicability in reliability engineering, materials science, and epidemiological modeling.

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